INTRODUCTION 7
not absolutely irreducible and/or where
A
is not a field. It seems unlikely that there
is any other way to capture these properties, without doing something essentially
equivalent to working inside the algebra
H[1r].
We have digressed again. Returning to chapters 3 and 4, if R is a commutative
ring with
!
E
R, then by a quaternion algebra over R we mean, roughly, a central
R-algebra spanned by elements {1, i,j, k} so that
i
2
E
R,
j
2
E
R, and ij = -ji = k.
(The reader will just have to cope throughout with the fact that
k
denotes both
the ground ring and an element of a quaternion algebra.) In general, we do not
require {
i, j,
k} to be linearly independent over R. For r, s E R, we denote by
H(r,
s; R)
the free module quaternion algebra over
R
with basis {1, i,j, k} and with
i
2
= r,
j
2
= s, ij = -ji =
k.
An example is the matrix ring M(2,
R)
=
H(1, 1;
R),
"h" (1 0). (0 1)
k (
0 1)
Wit 2
=
0 -1
'J
=
1 0 '
=
-1 0 .
If¢:
H+[7r]----)
R is a suitable homomorphism, we show that R
®
H[1r]
is a
H+[7r]
quaternion algebra over R. For example, if ¢(A2
-
4) =rand ¢(A2
+
B
2
+
C
2
-
ABC- 4) =
s
are units in R, then R
®
H[1r]
= H(r,
s;
R). Here, A= T(a), B =
H+[7r]
T(b),C = T(ab), where
{a,b}
generate
1r
as above. Whenever r,s E Rare units,
one has H(r, s; R') = M(2, R'), where R' is either R or a quadratic extension of R.
This provides another method in Chapter 4 for constructing absolutely irreducible
representations
p :
1r ----)
S£(2,
R')
via
H[1r] ----)
R
®
H[1r]
=
H(r, s; R)
~
H(r, s; R')
=
M(2, R').
H+[7r]
In Chapter 7, we extend this method to arbitrary finitely generated groups
1r.
The Addendum to Chapter 3 contains only a couple of technical results which
we don't need to describe here. On the other hand, the Addendum to Chapter 4
contains something completely different. We compute
H[1r]
and T
H[1r]
when
1r
is
the group of a two-bridge knot. Although we do this for any ground ring, the result
is easier to state if!
E
k.
Then
H[1r]
is a quaternion algebra over
H+[1r],
but not
quite a free module over
H+[1r].
In terms of suitable elements i,j,k
E
H-[1r],
we
have the following structure formulas.
H[7r]
+[ ] H+[1r] . +[
1.
H+[1r]
H
1r
ffi
(Q(I, J))
2
ffi
H
1r
J ffi
(Q(I, J)/
k[I,J]
(IQ(I, J)) [V4 +I+ J]
where Q(I, J)
E
Z[I,
J] is a polynomial with integer coefficients, and where
i
2
=
I,j
2
=
J
E
H+[1r]
and ij
=
-ji
=
k. The element J4 +I+
J
is the trace of a
meridian.
If K
is a field of characteristic
f.
2, then the conjugate classes of absolutely
irreducible representations
p :
1r ----)
S£(2, K) correspond to homomorphisms ¢ :
H+[1r]----)
K for which ¢(Q(I,
J))
=
0, ¢(/)f. 0. The polynomial Q(I,
J)
E
Z[I, J]
thus defines a plane curve of (mostly) irreducible representations of the two-bridge
knot group
71".
These curves have been studied extensively by Riley [21] and Burde
[2], perhaps with different variables. With our choice of variables, Q(O, J) with
J
=
z2 is the Conway polynomial of the two-bridge knot, which is a knot invariant.
Previous Page Next Page